University of Cape Town

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The copyright of this thesis vests in the author. No quotation from it or information derived from it is to be published without full acknowledgement of the source. The thesis is to be used for private study or noncommercial research purposes only.

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Published by the University of Cape Town (UCT) in terms of the non-exclusive license granted to UCT by the author.

Using the Movability Number to model local clear-water scour in rivers

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Benjamin Abban

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A thesis submitted in partial fulfilment of the requirements for

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the degree of Master of Science in Engineering

Department of Civil Engineering University of Cape Town

February 2007

Declaration

I, Benjamin Abban, understand the meaning of plagiarism and declare that all work in this dissertation, save for that which is properly acknowledged, is my own. Neither the whole work nor any part of it has been, or is to be submitted for another degree at this or any other university.

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I empower the University of Cape Town to reproduce for the purpose of research the contents as a whole or in part in any manner whatsoever.

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Benjamin Abban

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Date


11

Acknowledgements I first and foremost want to thank Associate Professor Neil Armitage for his supervIsIon, encouragement, support and faith in me during my studies at the University of Cape Town. I also

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want to thank Dr Chris Meyer, Margie Cunninghame, Alamgir Kabir, Nic Gibson, Andrew McBride, Noor Hassen and Elvino Witbooi for their respective contributions and support towards the research. Finally, I would like to thank my parents, Mr and Mrs JF Abban, and my sisters, Rosemary, Catherine and Rita, for the immense love and support they gave me throughout the study period.


iii

Abstract

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Local scour is associated with a considerable number of bridge failures worldwide. It occurs at bridge piers and abutments as a result of interactions between complex flow features and the channel bed. The number of factors involved in the interactions makes it difficult to predict. A lot of research has therefore been performed by several investigators to gain insight into the scouring process and scour prediction. Currently, local scour is estimated using physical models, empirical formulae or numerical models. Of these methods, the use of numerical models appears to be more economical and ideal as it permits flexibility in the choice of flow parameters and allows different scenarios to be easily studied. The aim of this research was thus to investigation into the use of the commercial CFD code FLUENT 6.2 for scour prediction based on the Movability Number.

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The research, which focused exclusively on local clear-water scour at bridge piers, stemmed from previous works performed by Armitage & McGahey (2003) and Cunninghame (2005) in which the Movability Number approach was developed and assessed. Results from these studies indicated that there was considerable potential in the Movability Number approach and, also, there was a need for a completely automated procedure for scour prediction based on the approach. For the current research therefore, an 'equilibrium model' was developed in which the river bed was successively modified in response to computed bed Movability Numbers until the final result reflected an equilibrium clear-water scour hole.

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Unstructured grids were generated in GAMBIT 2.2 and imported into FLUENT for the simulations. The symmetry condition was applied and the grids were fined up in regions where large velocity gradients or changes in other fluid properties were expected. Before the clearwater scour evolution simulations were carried out, the performance of the standard k-& model was compared with that of the Reynolds Stress model, and standard wall functions with nonequilibrium wall functions for a flat bed. Both turbulence models predicted similar scour patterns. Results of the numerical simulations were compared with data from a physical model and it was found that the non-equilibrium wall functions predicted scouring in regions on the bed where scour was not observed in the physical model. The standard wall functions, on the other hand, appeared to give realistic results. Since the standard k-&model involved the solution of two transport equations whilst the Reynolds stress model involved the solution of seven, the former was used with the standard wall functions for the scour hole evolution simulations. It was believed that this would result in shorter simulation times. A scour potential was defined as the difference between a computed bed Movability Number and the critical Movability Number required for sediment movement. Scour was considered to occur at those locations where the scour potential values were greater than zero and the grid nodes were displaced in response. User-defined functions were written to perform the bed modifications and ensure the integrity of the mesh as the bed geometry changed. Five physical scour experiments were simulated numerically. These physical experiments were performed as part of the research and were carried out in a 0.6lm wide tilting flume in the Hydraulics laboratory of the Civil Engineering Department at the University of Cape Town. Results from the numerical simulations were compared with those from the physical models. Using the Movability Number to model local clear-water scour in rivers

IV

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Although, the numerically estimated equilibrium scour depths were relatively close to those from the physical models, the shapes of the scour holes were not that similar. This was attributed to numerical difficulties in accurately predicting the flow field (and hence the Movability Numbers) at the bed. It was recommended that ways of improving the accuracy of the flow field prediction be found in order to accurately predict the bed Movability Numbers. In general, however, the Movability Number approach showed considerable potential for use in the prediction of local clear-water scour.


v

Table of contents Declaration Acknowledgements Abstract Table of contents List of figures List of tables List of symbols List of acronyms

1 11 111

v viii xi xii xvi

Chapter 1 Introduction

1-1

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Chapter 2 A review of local scour and associated phenomena 2.1 Shear stress distribution 2.2 Boundary layer 2.3 Velocity distribution for turbulent mean flows 2.4 Sediment properties 2.4.1 Sediment size Shape 2.4.2 2.4.3 Fall velocity 2.4.4 Particle size distribution 2.4.5 Cohesiveness 2.4.6 Angle of repose 2.5 Incipient motion 2.5.1 Forces on a cohesionless particle 2.5.2 Approaches to incipient motion Velocity approaches 2.5 .2.1 Shear stress approaches 2.5.2.2 2.5.2.3 Probability of pickup Stream power approach 2.5.2.4 2.5.2.5 The Movability Number approach 2.5.3 Incipient motion on sloping beds 2.6 Types of scour 2.6.1 General scour 2.6.1.1 Degradation, aggradation and regime conditions 2.6.1.2 Lateral channel migration 2.6.1.3 Bend scour 2.6.1.4 Confluence scour 2.6.2 Constriction scour 2.6.3 Local scour 2.6.3.1 Bed Features 2.6.3.2 Clear-water and live-bed scour 2.6.4 Temporal evolution of the scour hole 2.6.4.1 Controlling mechanism of scour Phases of scour hole development 2.6.4.2 Scour hole evolution 2.6.4.3 2.7 Summary Using the Movability Number to model local clear-water scour in rivers

2-1 2-1 2-3 2-5 2-9 2-9 2-9 2-10 2-11

2-13 2-13

2-14 2-14 2-15 2-15 2-18 2-19 2-22 2-24 2-26 2-28 2-28 2-28 2-29 2-29 2-29 2-29 2-30 2-31 2-33 2-35 2-35 2-36 2-37 2-39

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3-1 3-3 3-3 3-3 3-5 3-12 3-14 3-15 3-17 3-18 3-18 3-19

Chapter 4 Generation of physical data for model validation 4.1 Setup 4.2 Pathway of water 4.3 Flume setup 4.3.1 Scour region 4.3.2 Contamination and turbulence control 4.4 Measuring equipment 4.4.1 Flow measuring equipment 4.4.2 Scour measuring equipment 4.5 Experimental methods 4.5.1 False floors 4.5.2 Spreading sand 4.5.3 Establishing flow 4.5.4 Taking measurements 4.6 Criteria for flow selection 4.7 Summary of experiments 4.8 Scour hole evolution 4.9 Error estimation

4-1 4-1 4-2 4-3 4-3 4-4 4-6 4-6 4-7 4-8 4-8 4-9 4-11 4-12 4-12 4-15 4-17 4-18

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Chapter 3 Computational fluid dynamics in the modelling of local scour 3.1 Components ofa CFD model 3.1.1 Mathematical model 3.1.1.1 Governing flow equations 3.1.1.2 Turbulence modelling 3.1.2 Discretisation method 3.1.3 Coordinate system 3.1.4 Computational grid 3.1.5 Solution method and convergence criteria 3.2 CFD software 3.3 Model calibration and validation 3.4 Previous research on modelling oflocal scour at piers

Chapter 5 Procedures adopted for numerical modelling 5.1 Current research 5.1.1 Computational grids 5.1.2 Numerical solutions 5.1.2.1 Boundary conditions 5.1.2.2 Operating conditions 5.1.2.3 Models employed 5.1.2.4 Discretization schemes 5.1.2.5 Pressure-velocity coupling 5.1.2.6 Pressure interpolation 5.1.2.7 Under-relaxation 5.1.2.8 Solution initialization and convergence criteria Bed deformation model 5.1.3 Slope correction 5.1.4


5-1 5-1 5-1 5-3 5-4 5-6 5-6 5-10 5-11 5-12 5-12 5-13 5-13 5-16

vii

5.2

5.1.5 Accounting for angle ofrepose 5.1.6 Adapting the mesh Summary

5-18 5-19 5-21

Chapter 6 Results and discussions 6.1 Prediction of flow field 6.1.1 Grids 6.1.2 Free surface 6.1.3 Velocity field on flat bed 6.2 Scour prediction by different turbulence models 6.2.1 Flow fields and movability numbers 6.3 Modelling scour hole evolution 6.3.1 Mesh adaptation 6.3.2 Scour hole evolution

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Chapter 7 Conclusions

6-1 6-1 6-1 6-2 6-4 6-7 6-7 6-13 6-13 6-14

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Chapter 8 Recommendations for future research References

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Appendix


7-1

8-1 R-l A-I

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Figure

Description

Figure 2-1:

Uni-directional flow in an open channel

2-1

Figure 2-2:

Shear stress distribution through a column of flowing fluid

2-2

Figure 2-3:

Boundary layer along a smooth flat plat

2-4

Figure 2-4:

Relationship between Band k for uniform sand

2-6

Figure 2-5:

Velocity distribution - hydraulically smooth flows

2-7

Figure 2-6:

Velocity distribution - rough turbulent flows

2-8

Figure 2-7:

Velocity distribution - inner layer

2-8

Figure 2-8:

Chart for estimation of the fall velocity

Figure 2-9:

Cumulative and frequency distribution plots for particle size

2-11

Figure 2-10:

Angle 0 f repose of cohesionless sediment particles

2-13

Figure 2-11:

Forces acting on a river bed particle

2-14

Figure 2-12:

Sediment particles on river bed

Figure 2-13:

Chart for stable channel design

Figure 2-14:

Modified Shields diagram

Figure 2-15:

Influence of laminar and turbulent boundary flows on particle movement

Figure 2-16:

Probability of motion

2-21

Figure 2-17:

Distributions of stream power input and dissipation for open channel flow

2-22

Figure 2-18:

Flow pattern around a bridge pier

2-31

Figure 2.19:

Various types of bed features

2-32

Figure 2-20:

Migration of dunes and anti-dunes

2-33

Figure 2-21:

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List of figures

Variation of scour depth with velocity

2-34

Figure 2-22:

Variation of scour depth with time

2-34

Figure 2-23:

General scour patterns upstream of cylinder

2-35

Figure 2-24:

Evolution of scour hole

2-36

Figure 2-25:

Temporal evolution of scour depth for different flow intensities

2-38

Figure 2-26:

Temporal evolution of scour depth

2-38

Figure 3-1:

Relative cost of computing

3-1

Figure 3-2:

Velocity variation at a point in turbulent flow

3-5

Figure 3-3:

Finite Difference approximations for a given interval

3-12

Figure 3-4:

Control volumes - Finite Volume approach

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2-11

2-15 2-16 2-19 2-20

IX

Finite Element and Finite Volume approaches

3-14

Figure 3-6:

Cartesian grid

3-15

Figure 3-7:

Different types of grids

3-16

Figure 3-8:

Iterative solution procedure used in FLUENTs segregated solver

3-17

Figure 4-1:

Experimental setup

4-1

Figure 4-2:

Snapshots of scour region and platform setup

4-4

Figure 4-3:

Contamination control

4-5

Figure 4-4:

Turbulence control structure

4-5

Figure 4-5:

Weir in return channel

4-6

Figure 4-6:

Flow measuring devices

4-6

Figure 4-7:

Laser pointer rig

4-7

Figure 4-8:

Angles at which the extents of the scour hole were measured

4-7

Figure 4-9:

Periscope for scour depth measurement

4-8

Figure 4-10:

Upstream false floor indicating the support structure

4-9

Figure 4-11:

Sand spreader

4-9

Figure 4-12:

Placement and levelling of sand

Figure 4-13:

Scour results vrs Melville & Chiew's empirical formula

Figure 4-14:

Scour results vrs Ahmed's empirical formula

Figure 5-1:

Layout of computational domain

5-2

Figure 5-2a:

Horizonal section through computational grid

5-3

Figure 5.2b:

Vertical section through computational grid

5-3

Figure 5-3:

Wall treatment approaches

5-9

4-11

4-17 4-18

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Figure 3-5:

Vectors on a horizontal plane and a bed face

5-16

Longitudinal slope determination

5-17

Figure 5-6:

Adjustment of bed slope to reflect sliding

5-19

Figure 5-7:

Illustration of mesh adaptation

5-20

Figure 6-1 a:

Plan view of assessed meshes - near-pier region

6-1

Figure 6-1 b:

Plan view of assessed meshes - near-pier region

6-2

Figure 6-2a:

Depth contours of free surface

6-2

Figure 6-2b:

Depth contours of free surface

6-3

Figure 6-3:

Variation of water surface around pier

6-3

Figure 6-4:

Velocity vector fields in front of pier (along symmetry plane) and on bed

6-4

Figure 6-5:

Approach velocity distribution (0.3m upstream)

6-5

Figure 5-4: Figure 5-5:


x

Figure 6-6:

Downflow velocity distribution in front of pier

6-5

Figure 6-7 a:

Flow pattern in the lee of the pier

6-6

Figure 6-7b:

Flow pattern in the lee of the pier

6-7

Figure 6-8a:

Velocity fields predicted by different turbulence models in front of pier

6-8

Figure 6-8b:

Velocity fields predicted by different turbulence models in front of pier

6-9

Figure 6-9a:

Movability Numbers predicted by k-& model within the vicinity of the pier

Figure 6-9b:

6-10

Movability Numbers predicted by RSM within the vicinity of the pier

Figure 6-10:

6-11

Movability Numbers predicted by RSM with enhanced

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wall functions for Experiment 1

6-12

Illustration of mesh adaptation for different ratios

Figure 6-12:

Evolution of scour hole near the pier at 0°,30°,60° and 90° (physical model) 6-14

Figure 6-13:

Evolution of scour hole predicted by numerical model

6-15

Figure 6-14:

Movability Numbers at equilibrium scour hole

6-16

Figure 6-15:

Variation of maximum scour depth with boundary adjustment

6-16

Figure 6-16:

Contours of equilibrium scour holes from laboratory experiment

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and numerical model (Experiment 1) Distribution of shear stress with depth for different wall models

Figure 6-18:

Velocity vector fields at symmetry plane (upstream) and

6-17 6-18

6-18

Scour holes predicted by linear and exponential deformation models

6-19

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90° to the symmetry plane

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Figure 6-19:

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Figure 6-17:

6-13


Xl

List of tables Description

Page

Table 2-1:

Angle of repose of quartzitic sand

2-14

Table 4-1:

Flow rates and mean flow depths for modelled flows

4-15

Table 4-2:

Summary of physical experiments

4-16

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Table


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List of symbols

f FDer FDero g g

h ht hw I

k kr ks k/ kt Kd

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b bp Be Br Bs c cp Cr Cd C, Cdijf, CD, Ck Cs Cseour C1, C2 Cf.1" Cd, Cc2 d di ds dse d 15.9 d50 d84.1 d

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an A A+

Description Largest triaxial dimension of sediment particle nth interval Area normal vector Cross-sectional area of flow Damping constant Intermediate triaxial dimension of sediment particle Pier breadth Channel breadth Roughness coefficient Roughness coefficient for sand Shortest triaxial dimension of sediment particle Specific heat of fluid Face centroid vector Drag coefficient Turbulence model constants Constant dependent on wall roughness Coefficient in node displacement equation Turbulence model constants Turbulence model constant Sediment diameter Median diameter of particle size interval Maximum scour depth Equilibrium scour depth Sediment size for which 15.9% ofthe particles are finer Median particle diameter Sediment size for which 84.9% of the particles are finer Mean particle diameter Diagonal vector of bed face Unit vector in the vertical direction Roughness coefficient Diagonal vector of bed face Critical drag force for a given slope Critical drag force for a horizontal bed Acceleration due to gravity Gravity vector Depth of flow Fluid enthalpy Head of water above weir Intensity of motion Turbulent kinetic energy Roughness height Roughness height for sand Dimensionless roughness height Thermal conductivity Empirical expression for sediment coarseness

rs

Symbol a an


Unit m m m2 m2 m m m

m m

m

m m m m m m m m m m m N N mls2 mls2 m J m S-1

m 2/s 2 m m W/(mK)

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mqp

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p

r r

R Red Ref Reh Rek Rex Re. s

S Sf Saq SF t

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Pt PrE:> Prk

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P PI, P2 Pi Pr

Q

m mlm mlm

C

P Pi

q

Bed node vector Bed slope Friction slope Source term in continuity equation for phase q Shape factor Time

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Mn Mnc Mnlargest n nb ng N Na Ndns

m

n

m

Empirical expression for flow shallowness Empirical expression for flow intensity Empirical expression for time effects Mixing length Mixing length for inner layer Mixing length for outer layer Distance from the bed to half the mean flow depth Dissipation length scale Denotes maximum value Number of particle displacements within time t Mass transfer from fluid phase q to phase P Movability Number Critical Movability Number Largest Mn on the bed Manning's coefficient Manning's roughness for bed Manning's roughness for glass side walls Number of grid nodes on a vertical line Total number of surface particles over sample area Required number of grid points for direct numerical simulation Bed node vector Pressure Percentage by mass of particle size interval Bed node vector Wetted perimeter Pressure forces Stream power input at a point Power per unit volume required to keep particle in suspenSIOn Stream power dissipated at a point Turbulent Prandtl numbers Vector on bed face in the longitudinal direction Flow rate Ratio between two successive node intervals Bed node vector Hydraulic radius Pier Reynolds number Particle fall Reynolds number Reynolds number based on channel height Roughness Reynolds number Local Reynolds number Reynolds number based on shear velocity

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Kh KJ Kt I Ii 10 L Lm m


m m m m

kg

sm Jl3 sm Jl3 sm!!3

m N/m 2 m m N W/m 3 W/m 3

W/m 3 m

m3/s m m

s

XIV

u.

Time taken to achieve equilibrium Vector on bed face in the transverse direction Temperature Edge velocity Mean flow velocity at point P Shear velocity

u. c

Critical shear velocity

te t

T Ue up

Undisturbed approach velocity

u'" U

U Uc

v

VI

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v VI, V2, etc w w

W Xu

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x Y Ymin YP+ Y Yo Zo

~j

Ot Ot

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X 0

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a ap

P PI

o· AB At

AX ArjJ AY AYmax , AYmaxl G

Gijk

rjJ rjJmax rjJold

Point velocity in x direction Mean channel velocity Critical mean velocity Point velocity in Y direction Eddy viscosity Velocity vector Vortices Point velocity in z direction Vector in horizontal plane Weight of fluid element User specified constant Distance from leading edge Vertical distance from bed Vertical distance - lower limit of integration Distance from wall to point P Wall unit Distance from bed where velocity magnitude is zero Distance from wall where velocity magnitude is zero Under-relaxation factor Volume fraction of phase p in a cell Longitudinal slope Coefficient ofthermal expansion Function of Reynolds number Thickness of viscous sub-layer Kronecker delta Thickness of laminar boundary layer Thickness of turbulent boundary layer Displacement thickness Bed roughness factor Time step Grid spacing Computed change in rjJ Node displacement Maximum allowed node displacement Rate of viscous dissipation Permutation operator Arbitrary fluid property Maximum angle face makes with horizontal plane Old value of rjJ


s m K

mls mls mls mls mls mls mls mls mls Pa.s

mls mls m N m m m m m m

0

m m m m s m m m

m2 /s 3

0

xv

2 x 106 • Nalluri & Featherstone (2001) state that turbulence generally occurs at Rex = 5 x 105 . French (1994) suggests that the transition generally takes place between 5 x 105 and 106 . The change from laminar flow to turbulent flow occurs over a very short distance and this region is referred to as the transition region. The velocity gradients in the boundary layer reduce with increasing distance from the surface and the velocity approaches that of the main stream at the boundary layer fringes. In reality, there is no definite limit of the boundary layer. It is generally assumed that the boundary layer extends to the point at which the velocity is 99% of the main stream velocity (Liu, 1957; Massey & Smith, 1998; Chadwick et ai., 2004). The velocity distribution through the turbulent layer is also shown in Figure 2-3. The approximate thickness of the turbulent boundary layer, 8(, for Rex < 107 may also be determined from (White, 1991):

(2.9)

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Velocity distribution for turbulent mean flows

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The presence of roughness elements on the solid surface may have an effect on the boundary layer and flow in the main stream. If the main stream flow is laminar, roughness has no effect as the shearing action is solely due to viscosity throughout the fluid. In the case of turbulent flow, if the average height of roughness is smaller than the viscous sub-layer, the viscous sub-layer dampens the impact of the roughness and so there is little or no effect on the turbulent region. If the roughness elements protrude beyond the viscous sub-layer into the turbulent region, they will shed eddies and the turbulent component of shear stress will increase close to the surface (Nalluri & Featherstone, 2001; Chadwick et al., 2004).

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Yalin (1972) illustrates the derivation of the velocity distribution in the turbulent region of flow by representing the mixing length, I, in Prandtl's model with Ky and integrating Equation 2.6 from Ymin to y, where Ymin is the lower limit equal to the larger of the thickness of the viscous sub layer, 8, and the height of roughness, kr . Considering the relative size of the viscous sublayer thickness to the roughness height in the analysis, the following relationship for the velocity distribution is obtained: u 1 Y -=-In-+B u. K kr r

(2.10) Where u is the velocity at a depth Y and u. is the shear velocity given by:

(2.11)

Using the Movability Number to model local clear-water scour in rivers Chapter 2: A review oflocal scour and associated phenomena

2-6

The value of Br is dependent on the value of the roughness Reynolds number, Rek = u.kJv . Nikuradse (1933) determined the relationship between Br and Rek for equal size sand grains tightly glued on the boundary. Figure 2-4 indicates the relationship that was observed. The variables Br and kr are substituted with Bs and ks respectively for sand.

11

---

8 7

6

/

-/ -1-

/

i

Smooth

,

~

- --

--- --- ~ !

Transition

(

Completely rough

~

u.k, '" 70

u.ks '" 5

{5 o

----

Ii

I

0.2 0.4 0.6 0.8

1.0

1.2

1.4

1.6

1.8

I 2.0 2.2

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5

V

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-8.5 - --

-~

~

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/

2.4

2.6

2.8

109(

3.0

u:

s

3.2 )

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Figure 2-4: Relationship between Br and kr for uniform sand (after Nikuradse, 1933)

rs

Bs

= 2.51n

u:

s + 5.5 (2.12)

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For Rek < 5:

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From Nikuradse's (1933) experiment it was noted that

For Rek > 70:

Bs

= 8.5 (2.13)

According to Van Rijn (1993), turbulent flows corresponding to Rek < 5 are referred to as hydraulically smooth flows. These flows are characterised by the fact that the velocity distribution is independent on the size and nature of the roughness elements. This can be seen by substituting Equation 2.12 into Equation 2.10 and taking Kas 0.4. The result, Equation 2.14, is a relationship which is independent of roughness . y

.!!....=2.51n(u· )+5.5 u.

v

(2.14)

Using the Movability Number to model local clear-water scour in rivers Chapter 2: A review of local scour and associated phenomena

2-7

Fully developed turbulent flows or rough turbulent flows are those flows in which Rek > 70. The velocity in this type of flow is dependent on roughness and independent of molecular viscosity. The roughness elements are exposed to the turbulent region of flow, hence the term fully developed turbulent flow. Turbulent flow given by the condition 5 < Rek < 70 is considered to be in a transitional regime. The velocity distribution for a transitional regime is dependent on both the viscosity and roughness (Van Rijn, 1993). For hydraulically smooth flows, the relationship between the dimensionless velocity and the dimensionless depth within the viscous sub-layer is linear (Yalin, 1972) i.e.

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(2.15)

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where Re. = u.h/v. The equations for hydraulically smooth flows are also presented by Versteeg & Malalasekera (1995), Chanson (2004), Fluent (2005) etc. The velocity distribution for hydraulically smooth flows is shown in Figure 2-5. Both axes are represented in dimensionless quantities.

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r-----------------r-----=---

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Y h

Equation 2.15

1

o

k)h

u

u.

Figure 2-5: Velocity distribution - hydraulically smooth flows (after Yalin, 1972)

It can be deduced from Equations 2.10 & 2.13 that the velocity distribution for rough turbulent flow is dependent on the height of roughness. Substituting Br = 8.5 into Equation 2.10 results in a velocity distribution equation that is different from that of a hydraulically smooth flow. The distribution is however still logarithmic as shown in Figure 2-6.


2-8 y h

1~------------------~-----L-

n

Equation 2.10 (B r = 8.5)

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u

u.

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Figure 2-6: Velocity distribution - rough turbulent flows (after Yalin, 1972)

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It can be noted in Figure 2-5 that Yalin (1972) ignored the fact that there is a transitional region between the viscous sub-layer and the turbulent layer. Figure 2-7 is adapted from Schlichting & Gersten (2000). It illustrates the velocity profiles within the different sub-layers of the boundary

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layer. At present, there is no obvious relationship for the velocity distribution within the transitional regime. Some investigators simply assume that the viscous sub-layer and the turbulent region coincide at a value of u.y / v between 11 and 13 (Yalin, 1972; Rooseboom, 1992). Fluent (2005) suggest that 11.8 is a generally accepted value. Curves are also sometimes used to smoothly link the viscous sub-layer to the transition layer (Reynolds, 1974).

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u.y v

1

Outer layer

500 ~----------------------------~

Turbulent (log-law) layer Inner layer

30 - 70 Transitional layer

5

u Viscous (linear) layer

u,

Figure 2-7: Velocity distribution - inner layer (after Schlichting & Gersten, 2000)


2-9 The outer region occurs at u.y Iv > 500. The velocity distribution within this region is more dependent on the bed slope, flow depth, maximum flow velocity and large scale eddies instead of molecular viscosity and boundary roughness (Graf, 1998).

2.4

Sediment properties

2.4.1 Sediment size According to Julien (2002), the size of a sediment particle is its most important physical property. The size of a sediment particle, among others, dictates the ease with which it will be transported. Sediment size is easily defined and, depending on the definition being used, may be relatively easy to determine. Several definitions have been used to describe sediment size. These include (Yang, 1996; Raudkivi, 1998; Julien, 2002): Sieve diameter - this is the length of the side of a mesh square opening through which the sediment particle will just pass.

•

Sedimentation diameter - this is the diameter of a sphere with the same density and fall velocity as the particle in the same fluid and at the same temperature.

•

Nominal diameter - this is the diameter of a sphere with the same volume as the sediment particle and is usually determined by the volume the particle displaces.

•

Standard fall diameter - is the diameter of a sphere of specific gravity of 2.65 with the same fall velocity as the particle in quiescent distilled water of infinite extent at a temperature of

C

ap e

To w

n

•

Triaxial dimensions - these are lengths a, b and c along three mutually perpendicular axes through the particle such that a is the longest, b is intermediate and c is the shortest.

ity

•

of

24°C.

U ni

ve

rs

Generally, the definition used depends on the particle size e.g. the sieve diameter may be used for particles sizes ranging between cobbles and fine sand whereas the sedimentation or standard fall diameter may be used for sediment finer than fine sand (Chien & Wan, 1999).

2.4.2 Shape

The form which a particle takes is also important in the study of sediment transport. One common definition used to account for particle shape is given as follows (Yang, 1996; Raudkivi, 1998): a SF= ~ ",be

(2.16) where SF is the shape factor. Applying Equation 3.1, a perfectly spherical particle will yield a shape factor of 1. It has been determined that naturally worn quartz particles have an average shape factor of 0.7 (Yang, 1996). According to Armitage & McGahey (2003), the shape factor is used to take the deviation of a particle from the spherical into account as it is generally assumed that particles are reasonably spherical.


2-10

2.4.3 Fall velocity This is the terminal velocity that a particle attains in a quiescent column of water that is unbound. The fall velocity is dependent on particle size, shape, surface roughness, density, and viscosity and density of the fluid. The fall velocity, OJ, is also influenced by fluid boundaries, turbulence intensity levels and particle concentration (Yang, 1996; Raudkivi 1998). The expression for fall velocity is based on considerations of the balance between fluid drag and the submerged particle weight and is given as follows (Raudkivi, 1998; Chien & Wan, 1999):

OJ=

(2.17)

OJd v

Re = -

(2.18)

ap e

f

To w

n

where Ps is the sediment density, d is the sediment diameter, and Cd is the drag coefficient which is dependent on the particle fall Reynolds number defined as follows (Chien & Wan, 1999):

C

The change in Cd is small at Ref values greater than 1000. The fall velocity is thus considered to be independent of viscosity when Ref exceeds 1000 (Raudkivi, 1998; Chien & Wan, 1999).

U ni

ve

rs

ity

of

The fall velocity may also be read off a chart drawn up by the Sedimentation Subcommittee of the US Inter-Agency Committee on Water Resources. The chart relates the particle fall velocity to shape factor, temperature and sieve diameter and may be found in standard references (e.g. Yang, 1996; Raudkivi, 1998). It is shown in Figure 2-8.


2-11 10

~.....

I

'-"

~

!l.)

g

1.0

/I

,tI

,"

/I

'I

//

~r;

;;a

~

./ ~ ~

0

Z

?

, .'

0.1 0.06

v., f 0.2

, ./' ' ......

,:;.:

O"C 20"(40"C

~~ ;7 ~. .:.;;....

~

./ '

O"C 20"C 40"C

SF 0.5

I

0.1

/.'

~

.

~ if v

············24°C

..~.;;.:.o

,

.. ' ..'

O"C 20"C 40"C

I~I

1

~

1/

-,

10 I 1

1

SF 0.7

0.1 Fall velocity (cm/s)

100 I 10 I 1

I 100 I 10

n

,

~

·s

To w

«i

SF 0.9

I 50

2.4.4 Particle size distribution

ap e

Figure 2-8: Chart for estimation of the fall velocity (after Yang, 1996)

U ni

ve

rs

ity

of

C

The sizes of individual sediment particles in a group usually vary. As a result, sediment size is normally analysed using statistical methods. Parameters such as the mean size, standard deviation, skewness etc. are determined for a number of samples in order to obtain the characteristic properties of the group of particles. A common way of presenting sediment size is to plot a particle size-frequency distribution or a cumulative distribution curve as shown in Figure 2-9. In size-frequency distribution curves, particle size is divided into intervals and the percentage of the group falling within each interval is plotted as a function of size. Cumulative distribution plots typically show the percentage of particles finer than the individual sediment sizes (Raudkivi, 1998, Chien & Wan, 1999). 100

,-...

'lLre. Entrainment or a partidc may come ahout as a re,ull or an instantan~ous increa,e in the total drag, which CaUSeS the ei~ction of the pat1icle from its position ollthe b~d. An instantancous drop ill the local pressure may also r~sult in th~ ejeclioll of a pattic k du e to hydrostatic pressur~ ,

2.5.2 Approaches to incipient motion There are various models used to predict when motion will take place. \10st of them are relaled 10 a single now parameler that lS considered to playa dominanl role In predicling whether or not motion would occur, The three primary paramcters that arC used ollen arc velocity. hed sh~ar stress and str~am pow~r (Chicn & Wan, 1999). Approach~s based on these parameters arc d~scribed ill the followillg subsections. 2.5.2.1 Velncily apprnache, Vdocity approadles generally express incipient motlOn in tclTllS of eilhl"- the local or the mean stream vehx:ity, One ol'tlle early ,mdics on the maximum permissible mcan stream velocity tor

Using fh~ Mmvbiiil) Number w mOlld loml dear-'ml~r .",,!tIr in ri\'as Chapter 2: A ,,,jew of 1elaleJ pnelKJlnena

2-17 where u. c is the critical shear velocity based on d50 . It is given as follows: For O.lmm < d < 1mm: U. C

= O.OllS + 0.012Sdso I.4 (2.2Sa)

For 1mm < d < 100mm: U.

C

= 0.030Sdso O.s -

0.006Sdso -) (2.2Sb)

To w

n

Yang (1996) assumed a turbulent boundary and began his analysis with the local velocity. This local velocity was however integrated over the depth to obtain the average velocity. The final relationship therefore includes a dimensionless parameter based on the average velocity. Using Laboratory data from different investigators together with the theoretical considerations Yang (1996) presents the following formulae for incipient motion:

Vc

=

2.S + 0.66 log(u.d/v)-0.06

C

(j)

ap e

For 1.2 < u.d/v < 70:

of

For u.d/v ~ 70:

Vc

(2.26a)

= 2.0S (2.26b)

rs

ity

(j)

U ni

ve

According to Yang (1996), the relationships have been independently verified by Govers (1987), for laminar flow, and Talapatra & Gosh (1983), for turbulent flow. The use of the mean flow velocity as a criterion for incipient motion has also come under some criticism. According to Armitage (2002), the lift and drag forces are dependent on the velocity distribution and not the mean velocity. Raudkivi (1998) explains that the bed shear stress for a given mean velocity decreases with increasing depth of flow. This means that incipient motion will occur at higher mean velocities for relatively deeper channels compared to shallow channels with similar boundary conditions. The use of near-bed local velocity is thus a better option compared with the mean velocity. The problem with its use, however, is the fact that it is difficult to determine. The high velocity gradients near the bed and the irregular roughness elements make it difficult to define an elevation at which the velocity should be determined. Determination of the effective boundary location is made further difficult by changes in the nature of the boundary layer (Raudkivi, 1998).


2-18 2.5.2.2 Shear stress approaches

Shear stress approaches express incipient motion in terms of the bed shear stress. White (1940) derived the following relationship to determine the critical bed shear stress at which incipient motion would take place.

(2.27) where rc is the critical bed shear stress, d is the particle diameter, C is a constant dependent on the particle density and shape, and rf and rs are the specific weights of water and the particle respectively. The slope and the lift forces were considered to have an insignificant influence on incipient motion and so were neglected in its derivation (Equation 2.27).

ity

of

C

ap e

To w

n

The modified Shields relationship is perhaps the most popular shear stress approach to incipient motion. Shields in 1936 used dimensional analysis to derive a relationship which could be used to predict incipient motion. According to Shields (1936), the quantities important for incipient motion are the shear stress, r , the difference in density between sediment and fluid, Ps - Pf ' the particle diameter, d, the kinematic viscosity, v and the acceleration due to gravity, g. Shields (1936) neglected the effects of the lift force and obtained the following relationship:

(2.28)

U ni

ve

rs

Shields (1936) determined the relationship between the two quantities (LHS & RHS) experimentally using four different materials. These were amber (Ps = 1.06), brown coal (Ps = 1.27), granite (Ps = 2.7) and barite (Ps = 4.25). The product of the experiments was the well known Shields diagram, which indicates the threshold of movement. The modified Shields diagram is shown in Figure 2-14. It is based on work carried out by Shields (1936) and several other investigators. Shields (1936) originally did not fit a line to the experimental data. Extra data points and the line were added later based on experiments performed by other investigators.


2-19

Fully developed turbulent velocity profile 1.00

...

Turbulent boundary layer

I IIIIII

"I\.

I I I I I"

0.10

~

V threshold line

"

-

1'0..

~~

I'i"-o

0.02 1.0

10

100

1000

u.d v

ap e

0.2

-

n

i'

To w

1',

C

Figure 2-14: Modified Shields diagram (adapted from Yang, 1996, and Chien & Wan (1999)

U ni

ve

rs

ity

of

Shields diagram has corne under a lot of criticism by several researchers e.g. Yang (1996), Raudkivi (1998), Chien & Wan (1999) etc. In order to avoid the difficulty of determining the exact condition at which particles move, Shields (1936) measured values of T / d(ys - Yf) at least twice as large as the critical value and extrapolated to zero to obtain the values corresponding to no sediment discharge (Yang, 1999). According to Raudkivi (1998), the extrapolation introduces errors as transport rates at low values of shear stress excess, T - Tc' plotted on logarithmic paper do not have the same slope as those at high values of shear stress excess. A common criticism is that the axes are not independent of each other as the critical shear velocity appears on each. Trial and error is thus required to obtain a solution. According to Yang (1996), the rate of sediment transport cannot be uniquely determined by shear stress and so the use of shear stress as a criterion for incipient motion is questionable. 2.5.2.3 Probability of pickup

Figure 2-15 illustrates the influence of the type of boundary flow on particle movement (in the context of shear stress). As can be seen, the critical stress required for individual particle movements differ. When the near-bed flow is laminar, a steady shear stress is present. If this stress is larger than a particle's critical shear stress, it will move. A large shear stress will remove a greater number of particles.


2-20

,

~

C ................................................................. C

B .................................................................... B

-

A ............................................................... A

tLo

Flow

ap e

To w

n

Laminar Flow

.f . tf1~ ~IA r\:~ IV V

~ ...... ~~ ~V f\~ ,\~ i~ IIiJ ~i' l\j~

of

~~ ~~

II

ITO

Turbulent Flow

Flow

U ni

ve

rs

ity

.....

C

,

~

Figure 2-15: Influence of laminar and turbulent boundary flows on particle movement (Chien & Wan, 1999)

When the near-bed flow is turbulent, there are fluctuations in the applied shear stress and it is not clear which conditions will cause a particle to move. According to Chien & Wan (1999), even if a laminar sub-layer exists, strong eddies enter the layer from time to time generating fluctuations. It is clear that turbulent fluctuations and the differences in the force required for individual particle movement make it difficult to define a threshold above which there is particle movement. Many researchers have thus defined a 'pickup probability' (Armitage, 2002). This is based on the assumption of a normal distribution of the fluid forces applied to a particle. The distribution of the force required to move the particle is also considered to be normally distributed about some mean.


2-21

Probability Force required for movement

Fluid forces

No motion

Probability Force required for movement

ap e

To w

n

Fluid forces

ity

of

C

Incipient motion

Force required for movement

Fluid forces

U ni

ve

rs

Probability

General motion

Figure 2-16: Probability of motion (adapted from Van Rijn, 1993)

Figure 2-16 shows the probability density functions of the applied fluid forces and the force required for particle movement. Movement occurs only when there is an overlap of the two curves. The scale of movement will depend on the degree of overlap. A small overlap corresponds to the likelihood of a few particle movements on the bed. When the curve for the applied fluid forces completely exceeds the force required for movement, there is general particle movement over the entire bed surface.


2-22 Several definitions have been given for the intensity of particle motion (e.g. Kramer, 1935; Shields, 1936 etc.). Shvidchenko & Pender (2000a & 2000b) define the intensity of motion as follows:

(2.29) where I is the intensity of motion, m is the number of particle displacements during the time interval, t, and Na is the total number of surface particles over the sample area. In their definition, an I value of 10-4s-1 is considered as weak movement whereas 1= 10-2 s-l corresponds to general movement. These movements may be respectively described as occasional particle movement at some locations and frequent particle movement at many locations.

To w

n

2.5.2.4 Stream power approach

ity

of

C

ap e

This approach is based on the fact that a flowing fluid expends energy in moving a particle. Stream power is considered to be the rate of dissipation of fluid energy. Several investigators have applied stream power as a criterion for incipient motion. Armitage (2002) cites Bagnold (1960), Ackers & White (1973) and Yang (1972, 1973, 1976 & 1996) as examples. According to Armitage (2002), the problem with the approaches employed by these researchers is the fact that the stream power is expressed in terms of the mean velocity. The stream power is however proportional to the velocity gradient and thus varies across the channel depth. In order to be able to accurately describe incipient motion, the analysis must be carried out using local (near-bed) conditions.

U ni

ve

rs

Figure 2-17 shows the variation of the stream power input and the stream power dissipation for unidirectional flow through a channel.

Flow

Channel bed

Figure 2-17: Distributions of stream power input and dissipation for open channel flow (after Rooseboom, 1992)


2-23 For unidirectional flow, the point stream power input, 1>;, is given by:

1>; = pgSu (2.30) The stream power dissipated,

~,

at a point is given by: du dy

~ =Txy -

(2.31 )

To w

n

The distribution of stream power input is logarithmic as the point velocity profile is logarithmic (Section 2.3). As can be seen, the stream power input increases as one approaches the surface. This is in accordance with the velocity profile. The stream power dissipation, on the other hand, is greatest near the bed. This is because the shear stress and the velocity gradients are greatest near the bed (Armitage, 2002). It has been shown by White (1991) that the energy dissipated by a moving fluid (both in flowing

ap e

and transporting sediment) is as a result of shearing on the particle surfaces. The energy dissipated per unit volume, , referred to by Armitage & McGahey (2003) as the applied unit stream power, is thus expressed in indicial notation as follows (White, 1991):

C

au.

P ==T .. - ' t

ax. J

(2.32)

ity

of

lJ

U ni

ve

rs

The viscous stresses of an incompressible Newtonian fluid are proportional to the rates of deformation. They are therefore given by (White, 1991; Versteeg & Malalasekera, 1995):

Tij

au. au.]

=P - ' +_1 ax) ax;

(

(2.33)

Hence, if the velocity gradients are known, the applied unit stream power may be determined as follows:

(2.34) where Pb is the sum of both the laminar and turbulent viscosities.


2-24 Annitage (2002) analysed near-bed data from experiments carried out by several investigators and suggested the following criteria for incipient motion on a turbulent boundary: p

~~_r ,

Re. > 6.23

67

(2.35) where Pr is the power per unit volume required to keep a particle in suspension and is given by:

(2.36)

n

2.5.2.5 The Movability Number approach

ap e

To w

Liu (1957) noted that a requirement for ripple formation on a sediment-laden bed was that the flow had to be able to transport the sediment and, based on dimensional analysis and considerations of the scouring force of flow and sediment resistance, obtained the following criterion for sediment-ripple formation:

(2.37)

of

C

~ = f( U~d ,particleshape factor)

ve

rs

ity

The term u. / ()) was defined by Liu (1957) as the Movability Number (Mn). It has been shown by Rooseboom (1992) & Annitage (2002) that for incipient motion on a turbulent boundary, the value of Mn may be regarded as a constant. An expression for Mn on turbulent boundaries may be obtained as follows (Annitage, 2002):

U ni

Consider a rough turbulent flow. T t ~ To In the vicinity of the boundary. For a steady unidirectional flow in an open channel, To is given by: To

= PU. 2 (2.38)

Substituting 1= Ky into Equation 2.6, Prandtl's mixing length model may also be expressed as:

(2.39) For rough turbulent flows, therefore, we have:

(2.40)


2-25 Simplifying Equation 2.40 leads to the following expression for the velocity gradient:

au u.

-~

By KY (2.41) For unidirectional flow, the applied unit stream power may be obtained from Equation 2.32 as follows:

(2.42) 'xy

='t

~

'0 , substituting Equations 2.38 and 2.41 into Equation 2.42 yields:

(2.43)

ap e

To w

n

Since

of

C

The power per unit volume required to keep a particle in motion is given by Armitage & McGahey (2003) as follows:

ity

(2.36)

U ni

ve

rs

This is obtained from the fact that the power required to keep a particle in suspension is similar to the power dissipated if the particle falls through the fluid at terminal velocity. Particles moving along the bed are also considered to be in suspension but very close to the bed. The settling velocity of a sediment particle has been presented in Section 2.4.3 as:

0)=

3

p (2.17)

According to Chien & Wan (1999), a typical value of Cd for turbulent boundaries is 1.1. Substituting this value into Equation 2.17 and squaring both sides gives:

(2.44) According to Armitage & McGahey (2003), the power required to lift a particle from the bed should be greater than or equal to the power required to keep it in suspension. If the applied stream power required to dislodge a particle is considered to be directly proportional to the

Using the Movability Number to model local clear-water scour in rivers Chapter 2: A review of1oc~l scour and associated phenomena

2-26 power required to keep the particle in motion, then Equations 2.36 and 2.43 (Armitage & McGahey, 2003) give:

pu3

- ' oc (PS

Ky

- p)gw (2.45)

Dividing both sides by w 3 and applying Equation 2.44 yields:

3.3KY

u. 3 w

- 3o c - -

4d

(2.46)

(3.3K)~

U. -oc --

ap e

4

(2.47)

C

w

To w

n

As seen from Equation 2.46, the Movability Number varies with depth. lfthe critical value of the Movability Number required for incipient motion is determined at the distance y = d then Equation 2.46 becomes:

ity

of

This implies that for turbulent boundaries, the Movability Number required for incipient motion is a constant. This constant was estimated by Armitage (2002) to be 0.17 after analysing data from several investigators i.e.

=~I W

rs

Mn

=0.17,

critical

u.d > 6.23 V

(2.48)

ve

c

U ni

During the study, Armitage (2002) also estimated the relationship between the intensity of motion and the Movability Number to be: Mn

=

0.0066ln(I) +0.2405 (2.49)

2.5.3 Incipient motion on sloping beds The discussion on incipient motion has thus far concentrated on particle motion on a flat bed. It is to be expected that if the bed slopes downward in the direction of flow, a component of the particle's weight adds to the force that attempts to dislodge the particle. The result is that the particle is dislodged easier by a particular flow condition than it would have had it been on a flat bed. In terms of incipient motion criteria, the critical value would have to be multiplied by a certain factor, which is dependent on the slope, to reflect the relative ease in moving the particle (i.e. a smaller critical value is required for incipient motion). Similarly, if the slope is upwards it


2-27 becomes more difficult to dislodge a particle and a higher critical value is required for incipient motion. Two types of slope have been defined by Armitage & McGahey (2003) as follows: Longitudinal or Streamwise slope - this is the fall or rise of the bed in the direction of flow. A fall in the direction of flow is regarded as a positive slope and a rise as negative. Transverse slope - this is the fall of the bed in either direction normal to the direction of flow.

To w

FDer p (1 -tan pJ ---cos -FDerO tan (A

n

Van Rijn (1993) derived incipient motion criteria for both longitudinal and transverse slopes in terms of the drag force. If F Dcr is the critical drag force for a given longitudinal slope p, FDerO is the critical drag force for a horizontal bed and (A is the angle of repose, then for a longitudinal slope:

(2.50)

= cos r

(2.51 )

of

FDerO

C

F

--.l2£L

ap e

Similarly, if F Dcr is the critical drag force for a given transverse slope r then:

ve

rs

ity

Based on Equations 2.50 & 2.51, Armitage & McGahey (2003) present the following 'slope correction' factor for incipient motion criteria:

U ni

/fI

=

COSP(1- tanPJCOSr(1- tan: r JI/2 tan (A tan (A (2.52)

The slope correction factor may be applied to the Movability Number criterion for incipient motion as follows (Armitage & McGahey, 2003):

ul

-

O) erp,y

ul

-/fl-

0) erO

(2.53) Equation 2.49 was adjusted for the slope and relative roughness. The final equation for the intensity of motion was thus:

: = /fI [ 0.0066ln I

- 0.204

~ + 0.2405 ] (2.54)


2-28

2.6

Types of scour

Classification of the types of scour varies with literature. Raudkivi (1998), Breusers & Raudkivi (1991) and Melville & Coleman (2000) classify scour into 3 main categories, namely General, Constriction and Local scour. Hoffinans & Verhiej (1997) add Constriction scour to General scour and so have 2 main categories; General and Local scour. Richardson & Davis (2001) break the total scour at a highway crossing into Long Tenn Aggradation & Degradation, General and Local scour. May et al. (2002) classify scour into Natural, Constriction and Local scour and Graf (1998) also considers constriction scour under Local scour. It is therefore necessary to state clearly which classification is used in order to avoid any

ap e

To w

n

ambiguities in meaning. The classification used by Breusers & Raudkivi (1991), Raudkivi (1998) and Melville & Coleman (2000) will be used herein. Scour will be classified into General, Constriction and Local scour. Also, depending on the conditions under which scour develops, it may be considered to be either Clear-Water scour or Live-Bed scour (Breusers & Raudkivi, 1991; Hoffinans & Verhiej, 1997; Graf, 1998; Raudkivi, 1998; Melville & Coleman, 2000; Richardson & Davis, 2001; May et al., 2002).

2.6.1 General scour

U ni

ve

rs

ity

of

C

The various types of scour which are considered by Breusers & Raudkivi (1991), Raudkivi (1998) and Melville & Coleman (2000) as general may be categorized into the following: a. Degradation, aggradation and regime conditions b. Lateral channel migration c. Bend scour d. Confluence scour These types of scour are related to the river in general and depend mainly on the characteristics of the river and the catchment. They are not directly influenced by hydraulic structures and occur whether or not they are present. 2.6.1.1 Degradation, aggradation and regime conditions

Degradation and aggradation take place along the longitudinal direction of a river. They occur over a long period of time and result in changes in bed elevations along the longitudinal profile. They come about as a result of varying flow conditions and sediment supply (Richardson & Davis, 2001; May et at., 2002). When the rate at which scouring occurs exceeds the rate of deposition (long tenn basis), degradation is said to take place i.e. there is a deficit in sediment supply from upstream and this results in the lowering of the channel bed. On the other hand, if the deposition exceeds scouring, there will be a rise in the elevation of the channel bed and aggradation is said to take place. Regime conditions refer to the dynamic equilibrium conditions (flow and slope) to which a river has adjusted itself (May et al., 2002). Under such conditions, scouring and deposition occur at the same pace and hence aggradation or degradation does not take place. If the equilibrium conditions are disturbed by changes in water or sediment flows, net scouring or deposition is


2-29 likely to occur. Changes in flows may be due to human interference or natural changes such as urbanization, creation or removal of hydraulic structures, braiding of river channels, etc. 2.6.1.2 Lateral channel migration

This may be in the form of shifting bank lines, meander progression, movement of deep-water channels within channel banks or continuously changing channel positions in braided channels (Melville & Coleman, 2000; May et al., 2002). The channel migration may be gradual or rapid. Rapid migration usually occurs in response to flood events. A river is said to meander when it deviates from its straight course into a winding path. The meander progresses when, in the bends, there is continual scouring at the outer banks and deposition on the inner banks. This results in the movement of the entire channel.

n

2.6.1.3 Bend scour

ity

of

C

ap e

To w

This is scour that occurs in river bends as a result of the centrifugal force that acts on the water. According to Raudkivi (1998), this centrifugal force drives the water radially outward and results in the water surface possessing a super elevation. The super elevation of the water surface creates a lateral pressure gradient between the outer bank and the inner bank. Also, the magnitude of the centrifugal force decreases with depth due to decreasing velocity and results in a downward pressure gradient, which drives the water downward at the outer bank. The lateral pressure in tum drives the water inward toward the inner bank producing a kind of circulation in the cross-section. Flow goes outward at the top, downward at the outer bank, inward at the bed and upward at the inner bank. This circulation, together with the channel flow, produces a helical flow as the water moves along the bend. The helical flow decays a distance after the bend.

rs

2.6.1.4 Confluence scour

U ni

ve

A confluence is the place where two or more channels meet. Here, flows from the channels meet at different angles and their water levels, flow rates and slopes may also differ (Hoffmans & Verhiejj, 1997; Armitage & McGahey, 2003). When the streams of flow converge, secondary currents are induced i.e. lateral flows are produced resulting in helical flow patterns similar to the flow patterns obtained when two river bends are placed back to back. These helical flows usually result in the formation of deep scour holes with steep sides in the region of the confluence (Melville & Coleman, 2000).

2.6.2 Constriction scour This type of scour occurs in cross-sections of a river where the width is reduced. The contraction in width may be natural or as a result of the construction of a hydraulic structure such as a bridge (Melville & Coleman, 2000). Contraction of a channel leads to increased velocity of flow and shear stresses within the restricted section (May et aI., 2002). The consequence ofthese increases being the scouring of most or all of the bed at the section.


2-30

2.6.3 Local scour An obstruction placed in a flowing river changes the pattern of flow of the river in its neighbouring regions. Flow directions and pressure gradients are altered and often results in complex flow patterns. According to May et al. (2002), flow velocities and turbulence within the locality of the obstruction are increased. Depending on the nature of the obstruction, vortices may also develop. These increase the scouring potential of the stream in the locality of the obstruction. The result is the lowering of the bed in the immediate surroundings of the obstruction relative to the channel bed. This type of scour is referred to as local scour. Obstructions may be in the form of bridge piers, abutments, spur dikes, etc. Only flows around bridge piers are discussed here as they are the focus of the research. Figure 2.18 illustrates the complex flow features that develop around a bridge pier. The flow field has been described by several investigators. Armitage (2002) provides the following summary of the flow patterns:

To w

n

There is a rise in height of water at the leading face of the pier. This rise corresponds to the velocity head of the oncoming flow which is maximum at the surface. The flow is forced up and back onto itself to form a bow wave or "roller".

U ni

ve

rs

ity

of

C

ap e

There is a rise in pressure on the leading face of the pier (termed the "stagnation pressure '') and this increase decreases with the square of the velocity of the oncomingflow i.e. it decreases from top to bottom. This results in a partial reversal of the normal pressure gradient that provides the driving mechanism for a vertical "downflow". The maximum velocity of the downflow in vertical section, according to experimental data by Ettema (1980) and Raudkivi (1986), is situated between 0.05 and 0.02 times the pier breath, bp , upstream of the pier, being closer to the pier near the bed than at the surface. The velocity of the downflow also increases in magnitude towards the bed in the vertical section. If no scour is present, the maximum velocity is approximately 40% of the average oncoming velocity, U This velocity increases to approximately 0.8U as the scour depth increases past 2bp (Breusers & Raudkivi, 1991; Raudkivi, 1998). This relatively high velocity flow directed at the base of the pier acts as a sort of water jet that helps to initiate and maintain the scouring process (Oraf, 1998). The increase pressure on the leading face of the pier also helps to provide the necessary force for the acceleration of the flow around the sides. Once a scour hole begins to form, flow separation at the upstream rim results in the formation of a lee eddy that rotates in the opposite direction to the bow roller. This lee eddy is called the "horseshoe vortex" owing to its distinctive shape: it wraps itself around the front half of the pier and extends a few pier diameters downstream on either side before losing its identity and becoming part of the general turbulence. According to Raudkivi (1998), the vorticity of the horseshoe vortex is quite small and its main role in the scouring process comes about through its interaction with other flow structures. For example, it pushes the maximum downflow velocity within the scour hole closer to the pier. Flow separation around the sides of the pier results in the formation of "wake vortices" that alternately separate from the two sides to form a Von Karman vortex street. Near the bed, these vortices interact with the horseshoe vortex and, with their vertical low-pressure centres, lift the sediment from the bed like miniature tornadoes (Raudkivi, 1998).


2-31

Ilater ~ ur filc,::

\ .;h;lIlnd he.l

~

~L-

____- L - L_________

pit t

1r1l"'~1'\' " 1

ni

ve

rs

ity

of

C

ap

e

To

w

n

\

U

Figure 2 ~ IS: Flow pattel'" around:1 hddgc !Iier (arter Ralldkivi. L99S. and .\ ll'1yilll' & Colcman , 21100)

2.(i..1.1 Bcd Featurcs I oc,11 and cOl1 stri ction scour arc Ilsua ll y referred to ,IS IOCa li7ed scom (Mel vi lle & Coleman. :!()()()). Depending on lhe conditions lIlldel' whil:h loc;dizcu scour takes place. il may be cOlls id..:red to be ei the r clear-water sco ur or rive-hed scour. In order to undersfa nd live-bed SCOllr. it is necessary 10 fi rSI understand bCll fca lmcs ;md how they come about.

Bed ft"ahm.:s arc rel ie f fenlures initialed by the fl .. id o~i ll al i o n s generated downslream o f SOla II local ohstacles over a bottom consisti ng o f movabl e sediment l11:ltcrial s (Van Rijn. 1993 ). Thc-re arc gencrlllIy Number IU mudel {of/d d"(/"- IHU~'" ~CIiI Il I" /"il '~n

Chapler 2: 1\ review oflocnl osilc or ch imer
U

4.1

C ap

e

To w n

It

", rigu rc -I- I : Ex [lCrim cnta I srt u Jl (not d ra " n 10 sea Ie)

U'i"g IIw MOI'ubilily Num ix.'" Iv ~",.M local clcur-wulet . cow- m ril'ers CI~1Pte r 4; Cene""jon of ph, 'icnl dala ti:>< mode l "alidal>o"

..j.,2

Palhwa:y ol'watcr

The pathwll} that the \val"r 10rces ft"ln push ing It down the ch~J\Oel. !\

,ll1~11

U

Figu re 4-4: TUI'bu lcll cc cU lltr,,1 st r ucturc

U.ing II:" .I fmmhilin· .\·wn~,~· t" mnde! local dew'_watel" "r moool \'alida{ion

To w n

Figure -1_9: I'niscopc fo r sco nr delHI! measuremenT

Experimental methods

e

4.5

C ap

4.5. 1 False floors

ni v

er

si ty

of

rhe false floo rs wae erealed using pl ywoo d to allow 1;'1" the constrw;tion of the 8edim~nt h1l8in. Both l100rs \\~r~ at The Same l...:ighl. Tk heigh! ".a.~ selected 10 !Illo w l;lr approp lime scour depths 10 fully dC\Tlo p, f\ ramp. with a.u "'I'prosilllalt slo pe ,,1'1 :3. was lixed lO Ihe up81ream end o i" the upstream floor and t h~ flume bed, this \'as to provide tl sm(JOth lransilion n f' lhe water Irom the flum e hed onto the floor. A si milar ramp I\as placed at the do\,nstl"Cam end of lhe downstre!lm floor 10 ellSure lhat the water exi ted smooT hly from the lest region IlitholLt a ffecling the flo" depth. The length of the upstream floor lIas 6.1m and that nflhe dollnsTr~am floor WaS 1.91m_ Thes.: length s were 10 ensure the pro per d~,elopment of the required flow profile at the Test region.

U

The lil lse 11'"1"" were fixed to and support~d b} hollow substructul"Cs, Wlk:n the apptlftltus lIas initially run. some 01' the Wllk,- l10wed O\W tk>ors __ hi 1st some filled th~ spaces within the substruetul"C _ Ai r waS (hu s lnlpped jusl underneat h lhe 110l'wer ,en",. in ri''''"S ('hapTer 4; Generalion of ph)sicJl dala for model validation

To w n

Fig ure .t - W : [i l,strea m false nnnr indi c'lting fhr sU Pll'lrf ,fr ucf urr

e

45.2 Spre:l ding s:lnd

sand us~d for til 106 (See Section 2.2). Since the experiments were being performed for fully developed turbulent flows, the mean approach velocity, U, had to be (from Rex> 106 and Equation 2.8):

106 Xv x

U>---

(4.3) Taking x as the length of the platform (x = 6.1m) and substituting v with 1O-6m 2/s, the first criterion for flow selection was obtained as follows:

Q

Q

U ----->O 164 A 0.61h .

n

(4.4)

To w

where Q is the flow rate, A is the cross-sectional area of flow and h is the depth of flow.

ap e

The particle Reynolds number had to be considered for the experiments as well. For turbulent beds, the criterion developed by Armitage & McGahey (2003) requires that:

C

u.ks > 6.23 v

of

(4.5)

Substituting Equations 2.4 and 2.11 into Equation 4.5 and multiplying both sides by v / ks results

r::;:p 6.23v vghS > - -

ks

ve

rs

ity

m:

U ni

(4.6)

For uniform flows, the bed slope, S, may be replaced with the friction slope, Sf. The square root of the slope, SI/2, may be obtained from Manning's equation, as follows: 2

l..

nQp 3

S2 =--5-

A3 (4.7) Where p is the wetted perimeter and n is Manning's coefficient. Since the side of the walls and the bed are of different roughness, the equivalent Manning's coefficient may be determined from (Chow, 1959): (

Bc nb3/ 2 + 2hng3 / 2 )

2/3

(4.8)

Using the Movability Number to model local clear-water scour in rivers Chapter 4: Generation of physical data for model validation

4-14 where nb and ng are the Manning's roughness for the bed and the side walls respectively and Be is the channel width. For the glass side walls, ng was assumed to be 0.010 (Armitage, 2002). The value for nb was obtained from Strickler's relationship for shallow flows (Nalluri & Featherstone, 2001):

ks 1/6 (0.000725)1/6 n =-= =0012 b 26 26 . (4.9)

To w

n

Substituting Equations 4.7 and 4.8 into Equation 4.6 yields the following condition:

(4.10)

ap e

Substituting the appropriate values and simplifying resulted in the second criterion:

(4.11)

of

C

Q(O.OOI +0.002h)2/3 - = - - - - 2 - - - > 0.0012

U ni

ve

rs

ity

The flow rates and depths that were chosen for the experiments had to satisfy both Equations 5.4 and 5.11. Also, the flow conditions had to be such that the scour hole was not greater than O.2m. This constraint was due to the amount of space that was available in the recess (the depth of the recess). The third criterion was therefore:

d, r:':~=-.=.=. =--=-~;-==.====-=-.:(:--:==:,-\ - ---rVI chann~1

I

U

ni v

er

si ty

of

According to Fluent (::!OO5j, tit(' ]og:lrilhmic IiI'\" for mean vellICi\y is ,~]id for Ihe r~nge 30 < y ' < J()() for equi librium lurbukn l boundari~s_ In FLUF:-..rT (U, h(mever, the log~ri\hmic I:lw is applied whten y- > 11.225. C~1I si /~s w~r~ lhll~ st'lecl~d ~o lhmlheir cenlres "ould be clo,~ to but grealer lhan 11,225. This was ~~s~ary si ne~ lurbliJent houndary Ilo"s "ere ~ing moddleJ. It wou ld be poimkss to ~rform computatiom ill th~ laminar sub- I a>~r {y+ < 11 .225 j. The grid was fined up c1os~ to the pier, the hal and in the region "f the exp~eted fr~e surface. This was don~ so that the rapidly changing \clocity gradi~ms and flow fcatures could b~ ad equmel} resolved, The grid was generatcd in a manner such that it was IOughly aligned "ith lh~ e x~ded !low. This W,15 [ 0 limit f~lsc diffusion, ~ common nllmcric~l error Ih,11 lIC~llrs "hen the flow is nOlltli gned with the grid, False d i lTu~ion r~dllC~S \~ith reducing cell si/es. Although the cxpcCled d13nges in the hed elevation "ere small enough to suggest a very I,ne vertical mesh in the region ofthc surfac~, th~ kvd ofrcfinclllcnt was restricted by the aspcet ratio of the cells. Clre had [0 be tilken to prc\'em very high aspect ratios (excessive skewness) as this could le~d to prnbl~ms wilh solulion Cl~l\'ergencc, A finc \'ertical mesh would imply tlMt the mesh he fine in the hori/onlal plan.~ ~s welL Ihis wou ld, however. leild to a domilin with millicll1S or cells [hal would be compulation~lly ex~n\i\'~ lO soh e_ Ali & K:lrim (200::!) placed the inkt boondary and the outtlow boundary at J and 6 times Ihe pier radi us respe realisti~ and were in agreem ent with descriptions in literature (Rauukivi, I9ed nO\\ at thc base of thc pieL 'J his pallem was also noted in results presented b" OJsen & "vlclaecn (1993), Yen ct al. (2001), ,\li & Karim (2002) and Cunninghame (2005). Streamlines pl'esented by Raudkivi & Sut!\('rland (1982) for flow ut a ! and standmd k-f:models with hoth standard and non-~quilibl'iwn wall functions.

UJi"g 'he Mow,bili,y N~mber (0 mOilelloc-al d em'-wille' J CI'" o.-.",~; "",,'k' oM " ....,"'.......

h~

RSM

.... ~'" m ,.",,",,

6-12

To w n

As can be secn in Fil,.'lIrc 6-9 all the models predict an increa,e in Mn near the side wall s .cd in the physical cxperiment and ~o the lLSC nI' I'lmdions s~tl,itised to pressllre gradients (no ll-equ iliblium [unctions) did not app~,u to be reasonahle with in lh is !"gion. I[ow~\"er. non-equilibrium conditions ex ist within the picr vicini ty and the flow i~ inlllLetlCed to a large ~xtenl hy pressurc ~adiems. The appropriate method to usc close to the picr "ould thercfore he to Cfllplo\ nOn~qui l ihrium fllndions. Thc d"cision was made to try both the ~tandard and non-cquilibrium functions hy rcstric!ing hed de forma ti ons to the rcgionjust close to the pier. Unfol1unatdy_ the solution obtaincd for Experimcll1 [ predic led s~ollTin~ from the side wall to regions clos~ to the pier mld made it impossible to dcfine a rc,trided Mea "ithm " 'hieh deformation would he allow~d (s~e Fi~' lre 6-IO)./ln alternative solution would he to increasc th~ width ofth~ side channel to redll~~ lhe inllllcnce ofthc converging Ilow due to tl}!.; constriction, '['his approach would ho,,'cver reqllir~ exIra complll'llional power.

M"

C ap

e

0.1-' O,2U

_ _ Critic.1 Mn

of

0.16

si ty

0.1 2

U

ni v

er

0,08

Fil!U rr 6-1U:

'\Io~' ahili t~

wall

1\ u m her.< predicted by R!-o.' l

functi un ~

\I

ith

enhan~cd

fur r: ... pc r imcnt J

Il was also noted thal ),1oyability Numb~rs at tl}!.; first point of 110w separation upstream of thc picr were rdati,dy hlgh . The Mn Vlllucs wer~ quite close to the critical value implying thm scouring would have oc~llTTed in this region had lhe now been lilly strotlger. S
Vuml~·r OIer~:

fO model ioc
ni v

'"

•

U

35

,, "

• •

30

~

,t•

\"

er

"

/picr

00

of

Flo" d"ec!ioo

C ap

e

To w n

pr~dicted hy the modeL In th" bhorator).
;imum scour depths at lal'ious boundary adjustment steps, ' Jh~ cUrie is simi lar lO thal dc!.;imum seour depth asymptotkal lj.

1!.O5

,

---

• •

.

•

~

0,03

,i

0,02

#

U() l

"

/ -V·

.

AJjll'I"'""1

10

2.\

"

H gure 6-15: VaJ'iaJion of maximum

~coUl'

30

35

40

"

depth" il h boundary adju,tm ent

( i> ing Ihe .I10... "N/il), Sumner 10 model h:v/ clmr-wvler ,If,,,,'ahilily ,\'wllh", In mod,'lloral C{,,,Ir_wa!.>r scour ill I';WI',' "",,T I;' I.: jlt. ,j ; ....,," ;,..,.

6-19

U

ni v

er

si ty

of

C ap

e

To w n

Iltook long~r li)r th~ >cour hole to deve lop in th e casc of the exponential dclormation modcl

Extent, of predict ,d scour hole

Figure 6·19:

S~()ur

holes Jlrcdictcd b}'

lin~ar

and CXJloncntial ddormation modds

l;\mg [he ,\ /owN!i0' \'umi>L". to moJd loc al cI,ar_wUlel" ,'cow' :h 'r~- < 1,-""1,'"" ,,,'''"'

In

nv